Here is the full assignment:Consider three-point explicit schemes for the linear advection equation on the realline of the form$$u_{j}^{n+1}=u_{j}^n-C\left(u_{j}^n-u_{j-1}^n\right)+D\left(u_{j+1}^n-u_{j}^n\right) ., \tag{1}$$Show that$$\sum_j\left|u_j^{n+1}-u_{j-1}^{n+1}\right| \leq \sum_j\left|u_j^n-u_{j-1}^n\right|, \tag{2}$$
If $C \geq 0$, $D \geq 0$, and $C + D \leq1$
Model "solution":$$u_{j-1}^{n+1}=u_{j-1}^n-C\left(u_{j-1}^n-u_{j-2}^n\right)+D\left(u_j^n-u_{j-1}^n\right) ., \tag{3}$$
En soustrayant $\mathrm{Eq}(3)$ from $\mathrm{Eq}(1)$ we get$$u_j^{n+1}-u_{j-1}^{n+1}=(1-C-D)\left(u_j^n-u_{j-1}^n\right)+C\left(u_{j-1}^n-u_{j-2}^n\right)+D\left(u_{j+1}^n-u_j^n\right) ., \tag{4}$$
We take the absolute value of both sides, use the triangle inequality on the right-side as well as the fact that $|1-C-D|=(1-C-D),|C|=C$, and $|D|=D$, and then sum the two sides over all $j$ on we get:$$\sum_j\left|u_j^{n+1}-u_{j-1}^{n+1}\right| \leq(1-C-D) \sum_j\left|u_j^n-u_{j-1}^n\right|+C \sum_j\left|u_{j-1}^n-u_{j-2}^n\right|+D \sum_j\left|u_{j+1}^n-u_j^n\right| . \tag{5}$$
The right-side simplifies to give$$\sum_j\left|u_j^{n+1}-u_{j-1}^{n+1}\right| \leq \sum_j\left|u_j^n-u_{j-1}^n\right| , \tag{6}$$My question is the following
I didn't understand how do we get the inequality (6) ?
is $j$ start from - $\infty$ ?